Quillen’s Work on Rational Homotopy Theory

Abstract

In this chapter we review Quillen’s work on rational homotopy theory. Quillen gives a sequence of rational homotopy categories and proves that they are all equivalent. To a simply connected space, he associates a simplicial set with trivial one-skeleton and then the simplicial loop group. From there he passes to simplicial (complete) Hopf algebra, then to a simplicial Lie algebra, to a differential graded Lie algebra, and finally to a differential graded co-algebra. The latter dualizes to a differential graded algebra homotopy equivalent to the p.l. forms on the space, so that Quillen’s construction agrees in homotopy theory with Sullivan’s.